Some First-Order Probability Logics
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Theoretical Computer Science 247 (2000) 191–212 www.elsevier.com/locate/tcs View metadata, citation and similar papers at core.ac.uk brought to you by CORE Some ÿrst-order probability logics provided by Elsevier - Publisher Connector Zoran Ognjanovic a;∗, Miodrag RaÄskovic b a MatematickiÄ institut, Kneza Mihaila 35, 11000 Beograd, Yugoslavia b Prirodno-matematickiÄ fakultet, R. Domanovicaà 12, 34000 Kragujevac, Yugoslavia Received June 1998 Communicated by M. Nivat Abstract We present some ÿrst-order probability logics. The logics allow making statements such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. We describe the corresponding probability models. We give a sound and complete inÿnitary axiomatic system for the most general of our logics, while for some restrictions of this logic we provide ÿnitary axiomatic systems. We study the decidability of our logics. We discuss some of the related papers. c 2000 Elsevier Science B.V. All rights reserved. Keywords: First order logic; Probability; Possible worlds; Completeness 1. Introduction In recent years there is a growing interest in uncertainty reasoning. A part of inves- tigation concerns its formal framework – probability logics [1, 2, 5, 6, 9, 13, 15, 17–20]. Probability languages are obtained by adding probability operators of the form (in our notation) P¿s to classical languages. The probability logics allow making formulas such as P¿s , with the intended meaning “the probability of truthfulness of is greater than or equal to s”. Probability models similar to Kripke models are used to give semantics to the probability formulas so that interpreted formulas are either true or false. Every world from a probability model is equipped with a probability space. The corresponding probability measures are deÿned on sets of subsets of possible worlds. In this paper we present some ÿrst-order probability logics and explore their com- pleteness issues. The most general of our logics (denoted by LFOP1) is similar to the logic with probabilities on possible worlds [1, 9] which is appropriate for ana- lyzing degrees of belief. It is proved in [1] that no complete ÿnitary axiomatization is possible for that logic, and also that even its monadic fragment is undecidable. ∗ Corresponding author. E-mail: [email protected]. 0304-3975/00/$ - see front matter c 2000 Elsevier Science B.V. All rights reserved. PII: S0304-3975(98)00341-7 192 Z. Ognjanovic, M. RaskovicÄ / Theoretical Computer Science 247 (2000) 191–212 We give an inÿnitary axiomatic system for LFOP1 which we prove is sound and complete. In this paper the terms ÿnitary and inÿnitary concern meta language only. Object languages are countable, formulas are ÿnite, while only proofs are allowed to be inÿnite. We also put some restrictions on LFOP1 and investigate the resulting logics. The re- strictions are of the following kinds: only probability measures with ÿxed ÿnite range are allowed in models, only one probability measure on sets of possible worlds is allowed in a model, the measures are allowed to be ÿnitely additive, the set of mea- surable sets of possible worlds can be precisely described, etc. It is interesting that some of the logics obtained by the mentioned constraints have ÿnitary axiomatization, while some fragments are decidable. The rest of the paper is organized as follows. In Section 2 the most general of our logics is introduced, and its syntax, semantics and axiomatization are given. Deÿni- tions, formulations of statements and proofs from this section will be widely used in the remaining sections. In Section 3 we consider the case when only probabilities with ÿxed ÿnite range are allowed and give the corresponding axiomatization. In Section 4 an example which illustrates the relation between probability and modal logics is pre- sented. In Section 5 we consider the case when only one probability measure on sets of possible worlds is allowed. A further restriction of the ÿrst-order probability logics is given in Section 6 where some propositional probability logics are mentioned. In Section 7 the decidability of the considered logics is discussed and a decidable frag- ment of the probability logics is emphasized. We discuss some related papers in Section 8. 2. The logic LFOP1 In this section we present the logic LFOP1 (L for logic, FO for ÿrst-order, and P for probability). We describe its syntax and some classes of models, give an inÿnitary axiomatization and prove that it is sound and complete with respect to the mentioned classes of models. 2.1. Syntax The language L of the LFOP1-logic is an extension of the classical ÿrst-order language. It is a countable set which contains for each nonnegative integer k, k-ary k k k k relation symbols P0 ;P1 ;:::; and k-ary function symbols F0 ;F1 ;:::; and the logical symbols ∧, and ¬, quantiÿer ∀, a list of unary probability operators P¿s for every rational number s ∈ [0; 1], variables x;y;z;:::; and parentheses. The function symbols of the arity 0 are called constant symbols. Terms and atomic formulas are deÿned as in the ÿrst-order classical logic. The set of formulas is the smallest set containing atomic formulas and closed under forma- tion rules: if and ÿ are formulas, then ¬ , P¿s , ∧ ÿ and (∀x) are formulas. Z. Ognjanovic, M. RaskovicÄ / Theoretical Computer Science 247 (2000) 191–212 193 1 2 0 1 0 For example, the following is a formula: P¿s(∀x)P1 (x) → P3 (y; F0 ) ∧ P¿rP¿tP1 (F1 ). A formula is called classical if it contains no probability operators. In this section ;ÿ; ;::: are used to denote formulas. In a formula of the form (∀x) , is said to be the scope of that quantiÿer. An occurrence of a variable x in a formula is bound if it occurs in a part of which is of the form (∀x)ÿ. Otherwise, the occurrence is called free. A formula is a sentence if no variable is free in .If is a formula and t is a term, then t is said to be free for x in if no free occurrences of x lie in the scope of any quantiÿer (∀y), where y is a variable in t.If is a formula, and x1;:::;xm are variables, (x1;:::;xm) indicates that free variables of form a subset of {x1;:::;xm}.If (x) is a formula, and t is a term free for x in , then (t=x) denotes the result of substituting in the term t for all free occurrences of x. We will also use the shorter form (t) to denote the same substitution. We abbreviate: ¬(¬ ∧¬ÿ)by( ∨ ÿ), (¬ ∨ ÿ)by( → ÿ), ( ↔ ÿ)by(( → ÿ) ∧ (ÿ → )), ¬(∀x)¬ by (∃x) , ¬P¿s( )byP¡s( ), P¿1−s(¬ )byP6s( ), ¬P6s( )by P¿s( ), and P¿s( ) ∧¬P¿s( )byP=s( ). 2.2. Semantics We use the possible-worlds approach to give semantics to probabilistic formulas. It is similar to the objectual interpretation for ÿrst-order modal logics [7]. A class H of subsets of a nonempty set V is an algebra if it contains V and is closed under complementation and ÿnite union. A ÿnitely additive probability measure is a function from an algebra H to the real interval [0; 1] which satisÿes: (V )=1 and (H1 ∪ H2)=(H1)+(H2), for all disjoint sets H1;H2 ∈ H. An algebra H is a -algebra if it is closed under countable union. A function is a probability mea- sure if it maps a -algebra H to the real interval [0; 1] and satisÿes: (V )=1 and S∞ P∞ ( i=1Hi)= i=1 (Hi), for every disjoint sequence {Hi} of sets in H. An LFOP1-model is a structure M = hW; D; I; Probi where: • W is a nonempty set of objects called worlds, • D associates a nonempty domain D(w) with every world w ∈ W , • I associates an interpretation I(w) with every world w ∈ W such that: k k ◦ I(w)(Fi ) is a function from D(w) to D(w), for all i, and k, k k ◦ I(w)(Pi ) is a relation over D(w) , for all i, and k. • Prob is a probability assignment which assigns to every w ∈ W a probability space, such that Prob(w)=hW (w);H(w);(w)i, where: ◦ W (w) is a nonempty subset of W , ◦ H(w) is an algebra of subsets of W (w) and ◦ (w):H(w) → [0; 1] is a ÿnitely additive probability measure. Let M = hW; D; I; Probi be an LFOP1-model. A variable valuation v assigns some element of the corresponding domain to every world w and every variable x, i.e., v(w)(x) ∈ D(w). If w ∈ W , d ∈ D(w), and v is a valuation, then vw[d=x] is a valuation like v except that vw[d=x](w)(x)=d. 194 Z. Ognjanovic, M. RaskovicÄ / Theoretical Computer Science 247 (2000) 191–212 For a given LFOP1-model M = hW; D; I; Probi, and a valuation v the value of a term t (denoted by I(w)(t)v) is: • if t is a variable x, then I(w)(x)v = v(w)(x), and m • if t = Fi (t1;:::;tm), then m I(w)(t)v = I(w)(Fi )(I(w)(t1)v;:::;I(w)(tm)v). The truth value of a formula in a world w ∈ W for a given LFOP1-model M = hW; D; I; Probi, and a valuation v (denoted by I(w)( )v) is: m m • if = Pi (t1;:::;tm), then I(w)( )v = > if hI(w)(t1)v;:::;I(w)(tm)vi∈I(w)(Pi ), otherwise I(w)( )v = ⊥, • if = ¬ÿ, then I(w)( )v = > if I(w)(ÿ)v = ⊥, otherwise I(w)( )v = ⊥, • if = P¿sÿ, then I(w)( )v = > if (w){u ∈ W (w): I(u)(ÿ)v = >}¿s, otherwise I(w)( )v = ⊥, • if = ÿ ∧ , then I(w)( )v = > if I(w)(ÿ)v = >, and I(w)( )v = >, otherwise I(w)( )v = ⊥, and • if =(∀x)ÿ, then I(w)( ) = > if for every d ∈ D, I(w)(ÿ) = >, otherwise v vw[d=x] I(w)( )v = ⊥.